A self-adaptive moving mesh method for the short pulse equation via its hodograph link to the sine-Gordon equation

Abstract

The short pulse equation was introduced by Schaefer--Wayne (2004) for modeling the propagation of ultrashort optical pulses. While it can describe a wide range of solutions, its ultrashort pulse solutions with a few cycles, which the conventional nonlinear Schroedinger equation does not possess, have drawn much attention. In such a region, existing numerical methods turn out to require very fine numerical mesh, and accordingly are computationally expensive. In this paper, we establish a new efficient numerical method by combining the idea of the hodograph transformation and the structure-preserving numerical methods. The resulting scheme is a self-adaptive moving mesh scheme that can successfully capture not only the ultrashort pulses but also exotic solutions such as loop soliton solutions.